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December  2016, 21(10): 3637-3654. doi: 10.3934/dcdsb.2016114

Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays

Thanh-Anh Nguyen 1, , Dinh-Ke Tran 2, and  Nhu-Quan Nguyen 3,

1. 

Vietnam Education Publishing House, 81 Tran Hung Dao, Hanoi, Vietnam
2. 

Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam
3. 

Department of Mathematics, Electric Power University, 235 Hoang Quoc Viet, Hanoi, Vietnam

Received  November 2015 Revised  August 2016 Published  November 2016

We deal with the Cauchy problem associated with integro-differential inclusions of diffusion-wave type involving infinite delays. Based on the behavior of resolvent operator associated with the linear part, an explicit estimate for solutions will be established. As a consequence, the weak stability of zero solution is proved in case the resolvent operator is asymptotically stable.
Keywords: measure of non-compactness, Weak stability, MNC estimate., fixed point theory, condensing map.
Mathematics Subject Classification: Primary: 35B35, 37C75; Secondary: 47H08, 47H1.
Citation: Thanh-Anh Nguyen, Dinh-Ke Tran, Nhu-Quan Nguyen. Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3637-3654. doi: 10.3934/dcdsb.2016114
References:
[1]

B. de Andrade and C. Cuevas, $S$-asymptotically $\omega$-periodic and asymptotically $\omega$-periodic solutions to semi-linear Cauchy problems with non-dense domain, Nonlinear Anal., 72 (2010), 3190-3208. doi: 10.1016/j.na.2009.12.016.  Google Scholar

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T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006.  Google Scholar

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T. A. Burton and T. Furumochi, Fixed points and problems in stability theory for ordinary and functional differential equations, Dyn. Syst. Appl., 10 (2001), 89-116.  Google Scholar

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J. P. Carvalho dos Santos and C. Cuevas, Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations, Appl. Math. Lett., 23 (2010), 960-965. doi: 10.1016/j.aml.2010.04.016.  Google Scholar

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E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discrete Contin. Dyn. Syst., (Supplement) (2007), 277-285.  Google Scholar

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J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in $L^{1}(\mu, X)$, Proc. Amer. Math. Soc., 118 (1993), 447-453. doi: 10.2307/2160321.  Google Scholar

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R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, New York Inc., 1977.  Google Scholar

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A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Translated from the Russian. Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

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Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math., 27 (1990), 309-321.  Google Scholar

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C. Gori, V. Obukhovskii, M. Ragni and P. Rubbioni, Existence and continuous dependence results for semilinear functional differential inclusions with infinite delay, Nonlinear Anal., 51 (2002), 765-782. doi: 10.1016/S0362-546X(01)00861-6.  Google Scholar

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J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

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Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, vol. 1473, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.  Google Scholar

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M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, in: de Gruyter Series in Nonlinear Analysis and Applications, vol. 7, Walter de Gruyter, Berlin, New York, 2001. doi: 10.1515/9783110870893.  Google Scholar

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F. Mainardi and P. Paradisi, Fractional diffusive waves, J. Comput. Acoustics, 9 (2001), 1417-1436. doi: 10.1142/S0218396X01000826.  Google Scholar

[23]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[24]

R. Metzler and J. Klafter, Accelerating Brownian motion: A fractional dynamics approach to fast diffusion, Europhys. Lett., 51 (2000), 492-498. doi: 10.1209/epl/i2000-00364-5.  Google Scholar

[25]

J.-S. Pang and D. E. Stewart, Differential variational inequalities, Math. Program. Ser. A, 113 (2008), 345-424. doi: 10.1007/s10107-006-0052-x.  Google Scholar

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[27]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447. doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

[28]

T. I. Seidman, Invariance of the reachable set under nonlinear perturbations, SIAM J. Control Optim., 25 (1987), 1173-1191. doi: 10.1137/0325064.  Google Scholar

show all references

References:
[1]

B. de Andrade and C. Cuevas, $S$-asymptotically $\omega$-periodic and asymptotically $\omega$-periodic solutions to semi-linear Cauchy problems with non-dense domain, Nonlinear Anal., 72 (2010), 3190-3208. doi: 10.1016/j.na.2009.12.016.  Google Scholar

[2]

N. T. Anh and T. D. Ke, Decay integral solutions for neutral fractional differential equations with infinite delays, Math. Methods Appl. Sci., 38 (2015), 1601-1622. doi: 10.1002/mma.3172.  Google Scholar

[3]

E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001. Google Scholar

[4]

T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006.  Google Scholar

[5]

T. A. Burton and T. Furumochi, Fixed points and problems in stability theory for ordinary and functional differential equations, Dyn. Syst. Appl., 10 (2001), 89-116.  Google Scholar

[6]

J. P. Carvalho dos Santos and C. Cuevas, Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations, Appl. Math. Lett., 23 (2010), 960-965. doi: 10.1016/j.aml.2010.04.016.  Google Scholar

[7]

N. M. Chuong, T. D. Ke and N. N. Quan, Stability for a class of fractional partial integro-differential equations, J. Integral Equations Appl., 26 (2014), 145-170. doi: 10.1216/JIE-2014-26-2-145.  Google Scholar

[8]

C. Cuevas and J. César de Souza, Existence of $S$-asymptotically $\omega$-periodic solutions for fractional order functional integro-differential equations with infinite delay, Nonlinear Anal., 72 (2010), 1683-1689. doi: 10.1016/j.na.2009.09.007.  Google Scholar

[9]

C. Cuevas and J. César de Souza, $S$-asymptotically $\omega$-periodic solutions of semilinear fractional integro-differential equations, Appl. Math. Lett., 22 (2009), 865-870. doi: 10.1016/j.aml.2008.07.013.  Google Scholar

[10]

E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discrete Contin. Dyn. Syst., (Supplement) (2007), 277-285.  Google Scholar

[11]

J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in $L^{1}(\mu, X)$, Proc. Amer. Math. Soc., 118 (1993), 447-453. doi: 10.2307/2160321.  Google Scholar

[12]

R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, New York Inc., 1977.  Google Scholar

[13]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.  Google Scholar

[14]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Translated from the Russian. Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[15]

Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math., 27 (1990), 309-321.  Google Scholar

[16]

Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation. II, Osaka J. Math., 27 (1990), 797-804.  Google Scholar

[17]

C. Gori, V. Obukhovskii, M. Ragni and P. Rubbioni, Existence and continuous dependence results for semilinear functional differential inclusions with infinite delay, Nonlinear Anal., 51 (2002), 765-782. doi: 10.1016/S0362-546X(01)00861-6.  Google Scholar

[18]

J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.  Google Scholar

[19]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[20]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, vol. 1473, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.  Google Scholar

[21]

M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, in: de Gruyter Series in Nonlinear Analysis and Applications, vol. 7, Walter de Gruyter, Berlin, New York, 2001. doi: 10.1515/9783110870893.  Google Scholar

[22]

F. Mainardi and P. Paradisi, Fractional diffusive waves, J. Comput. Acoustics, 9 (2001), 1417-1436. doi: 10.1142/S0218396X01000826.  Google Scholar

[23]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[24]

R. Metzler and J. Klafter, Accelerating Brownian motion: A fractional dynamics approach to fast diffusion, Europhys. Lett., 51 (2000), 492-498. doi: 10.1209/epl/i2000-00364-5.  Google Scholar

[25]

J.-S. Pang and D. E. Stewart, Differential variational inequalities, Math. Program. Ser. A, 113 (2008), 345-424. doi: 10.1007/s10107-006-0052-x.  Google Scholar

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[27]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447. doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

[28]

T. I. Seidman, Invariance of the reachable set under nonlinear perturbations, SIAM J. Control Optim., 25 (1987), 1173-1191. doi: 10.1137/0325064.  Google Scholar

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